Because there are children involved, my advice to you would be to break off the affair...and for God's sake DO NOT TELL YOUR GIRLFRIEND!!! Talk to a counsellor, and see if there is a chance to salvage your current relationship with some work on your part (and hers, obviously, because you wouldn't be cheating all the time if you were entirely happy).
If not for the children involved, I'd advise you to walk, but it's just not that simple in this instance...you owe it to your children to at least try to fix the broken relationship. If you give it an honest effort, and you're still not happy, then by all means break up and move on.